Mossbauer Spectroscopy

Emily P. WangMIT Department of Physics

The ultra-high resolution (∆EE

= 10−12) method of Mossbauer spectroscopy was used to probevarious nuclear effects. The Zeeman splittings of the ground and first excited state of iron-57 werecalculated to be spaced apart by (0.8± 0.2)× 10−7eV and (2.0± 0.3)× 10−7eV, respectively. Thetheoretical values are 1.1× 10−7 and 1.9× 10−7. The ratio between the nuclear magnetic momentsof the first excited state and the ground state of the iron-57 atom was calculated to be −1.2± 0.3,while the value from literature [1] is −1.715 ± 0.004. The quadrupole splitting of the 3d6 stateof Fe++ was measured to be (2.1 ± 0.5) × 10−7eV, while the accepted value is 1.5 × 10−7eV.The isomer shift in Fe++ was measured to be (1.7 ± 0.5) × 10−7eV, while the accepted value is6.7×10−8eV, and the isomer shift in Fe+++ was measured to be(8±6)×10−8eV, while the acceptedvalue is 2.4 × 10−8. The resonant line width was measured to be 6.8 ± 17.1 × 10−8eV , while theaccepted value is 4.7 × 10−9eV . The lifetime of the first excited state of iron-57 was calculated tobe (6.7 ± 2.6) × 10−9seconds, while the accepted value is 9.8 × 10−8 seconds. The large errors inthe resonant line properties can be attributed to the inaccuracy in absolute calibration and error indrive velocity characteristics.

1. INTRODUCTION

The phenomenon of fluorescence was first observed inthe mid-1800s by Stokes. Stokes observed that solids,liquids, and gases placed under certain conditions wouldpartially absorb incident electromagnetic radiation thatwas then re-radiated. A special case of this phenomenonis resonance fluorescence, which occurs when the inci-dent and the re-emitted radiation are of the same wave-length. Scientists were at first prevented from observingresonance absorption on a nuclear level due to the recoil-energy loss of the re-emitted quantum, which caused theemission line to be shifted away from the transition en-ergy required for resonant absorption to occur. Success inobserving resonance absorption was eventually obtainedthrough various methods that made use of Doppler shift-ing to counter the energy shift caused by the recoil. In1960, Rudolph Mossbauer discovered a method of car-rying out “recoilless” spectroscopy by locking the targetnuclei into a crystal lattice [2]. Because the crystal lat-tice’s phonons have an extremely low probability absorb-ing any recoil energy from the incident radiation, all ofthe excitation energy is transferred to the emitted quan-tum. These conditions allowed for resonance absorption.

Mossbauer spectroscopy has made possible the probingof the hyperfine structure of an atom, which normally re-mains obscured by gamma lines, since the thermal widthof a gamma line is large in comparison to the spacing ofthe hyperfine levels of a nucleus. Sharper nuclear tran-sitions, the 14.4keV transition in 57Fe in particular, canbe used to measure many important effects [2].

2. THEORY

2.1. Recoil-Free Spectroscopy

The resonant scattering of optical photons has beenwell-known for many years. The classic example is that ofthe excitations of sodium vapor, a transition which findsa practical use in fluorescent lighting. In this case, thelight quanta emitted by atoms of the emitter, which thetransition from their excited states to their ground statesare also used to cause the opposite effect in the atomsof an absorber, which consists of identical atoms. Theatoms of the absorber undergo a transition from theirground state to their excited states, dropping back downwith the emission of fluorescent light [3].

The analogous process in nuclear gamma rays cannotbe immediately observed in the same fashion, due to thefact that the gamma ray loses energy to the recoil of thenucleus, causing the emitted gamma ray to have too littleenergy to be resonantly reabsorbed, since the quantummust provide the kinetic recoil energy in addition to thetransition energy [2]. Techniques involving the Dopplershift were used to compensate for this kinetic recoil en-ergy and return the system to resonance conditions.

In 1957, Mossbauer discovered a way of eliminatingthe problem of recoil by taking advantage of the factthat atoms locked into a crystal lattice cannot recoil ar-bitrarily. This fact is especially important in regimeswhere the normal, free-atom recoil energy is compara-ble to the energy of the quantized lattice vibrations, orphonons. When these conditions exist, it is highly prob-able for zero-phonon processes to occur. In that case,all of the energy of the incoming gamma ray goes intothe nuclear transition, and the recoil momentum is takenby the entire crystal [3]. Mossbauer’s method of probingnuclear structure has the unique feature that it insuresthe complete elimination of energy loss [2].

2

m = -3/2j

-1/2

1/23/2

-1/2

1/2

I=3/2

I=1/2

FIG. 1: Diagram of Zeeman splitting and the allowed transi-tions between the magnetic substates. The arrows indicatethat the sample’s Fe57 electrons in the lower energy sub-states are excited to higher energy substates when they collidegamma rays of the required energy. The remaining spectrumof gamma rays observed by the detector on the other side ofthe sample shows absorption peaks at the energies equal tothe energies of transition.

2.2. Zeeman Effect in Fe57

When at atom is placed into a magnetic field, its energylines are split into magnetic substates. This is known asthe Zeeman effect. In the Fe57 atom, the interactionbetween the nuclear magnetic moment and the internalmagnetic field of the atom allows us to observe Zeemansplitting. The radiation emitted or absorbed in the tran-sition between the ground and 14.4-keV level of Fe57 hap-pens to exhibit a clear example of a pure nuclear Zeemaneffect. This is due to the fact that there are no additionalquadrupole shifts in the nuclear energy levels, due to thecubic symmetry of the iron lattice [1].

The following equation relates g, the Lande constant,to µ, the magnetic moment of the nucleus, and B, theinternal magnetic field of the atom: g1 =

′mu1BI1

Theequation above gives the relation for the ground stateof the Fe57 atom, and there is an analogous equationdescribing the relation for the first excited state of Fe57.

2.3. Resonant Line Width

Due to the Heisenberg uncertainty principle, there isan uncertainty in the energies of the individual excitedstates of the nuclei, therefore the lines produced by tran-sitions between an excited state and the ground state willhave a certain minimum width. This width Γ is calledthe natural width, and is connected with the lifetime τ ofan excited state through the relation Γ = ~×ln2

Γ . Typi-cal values for the lifetimes of lower excited nuclear statesrange from 10−7 to 10−11 seconds, which means the nat-ural line widths for ground-state transitions range from10−8 to 10−4eV [2].

2.4. Quadrupole Splitting and Isomer Shift

Quadrupole splitting is a phenomenon that is causedby a nonsymmetric nuclear charge distribution. A nu-cleus that has a spin quantum number I that is greaterthan 1/2 will have a non-spherical charge distributionwhose magnitude of deformation Q can be expressed aseQ =

∫ρr2(3cos2θ−1)dV , where e is the charge of a pro-

ton, ρ is the charge density in a volume element dV at adistance r from the center of the nucleus and at an angleθ from the nuclear spin quantization axis [4]. Quadrupolesplitting serves to lift the degeneracy in the excited statesof a nucleus, and the magnitude of the splitting is givenby: ∆E = eQ

3m2I−I(I+1)

4I(2I−1)∂2φ∂z2 , where φ is the electric po-

tential and mI is the magnetic quantum number of thenuclear state. Quadrupole splitting is not seen in the I=1/2 ground state of Fe57, but it is seen in the I= 3/2state, and is given by the expression ∆E = ± 1

8Qe∂2V∂z2

Isomer shift is a nuclear effect that arises from a changein chemical environment between the source and the ab-sorber used in the Mossbauer setup. It involves the in-teraction between the nuclear charge and the electroniccharge within the nuclear volume–specifically, the s elec-trons, which have a finite charge probability density atthe origin, which is where the nucleus of the atom is lo-cated. The amount of isomer shift depends on the totalelectronic charge at the nucleus, which is different in eachelement [5]. The isomer shift can be calculated with thefollowing equation: ∆E = 2π

5 Ze2(R2

is − R2gs)(|ψ(0)a|2 −

|ψ(0)e|2), where Z is the nuclear charge, Ris is the ra-dius of the excited electronic state, Rgs is the radius ofthe ground electronic state, ψ(0)a is the electron wavefunction of the absorber, ψ(0)e is the electron wave func-tion of the emitter at R = 0.

3. EXPERIMENT

3.1. Michelson Calibration

In order to convert channel (bin) numbers into a mean-ingful value–namely, the velocity of the drive motor atany given time–it was necessary to do an absolute cali-bration. This was done with a Michelson interferometer,as shown in Figure 2. The interferometer functioned byrecording the interference between different laser beamsthat reached the photodiode. Interference was caused bythe fact that one of the mirrors in the interferometer wasattached to the moving drive, thus altering one of thepaths taken by the laser. As the drive moves towardsand away from the absorber, the photodiode sees a sinu-soidal wave that represents the constructive and destruc-tive interference due to the change in path length of thelaser beam. When the photodiode signal goes throughone cycle of constructive and destructive interference,this means that the drive has traveled a total distanceof λ/2, where λ is the wavelength of the laser. It is λ/2

3

HeNe Laser

C = NT(2Vi/λ)

λVi=C /2NT

Beam Splitter

Photodiode

Fixed Mirror

ProportionalCounter

Drive Motor

Path of Laser

Low-FrequencyAmplifier

FIG. 2: Setup of the Michelson calibration.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Channel Number

Cor

resp

ondi

ng V

eloc

ity (

cm/s

)

Michelson Calibration

Data

Right Side Fit

Left Side Fit

yleft

=(1.19±.01)−(.0012±.0001)x, χ2ν−1

=0.6

yright

=(−1.08±.01)+(.0010±.00001)x, χ2ν−1

=0.6

FIG. 3: Calibration data converted to velocity correspondingto each channel, with fit-lines drawn and fit equations shown.

and not lambda because the laser beam is reflected at thedrive mirror, so a given displacement actually increasesthe laser path length by twice that displacement. Theraw calibration data can be converted from the numberof counts in each channel into the drive velocity corre-sponding to each channel by using the equation Vi = Ciλ

2NTwhere Vi is the velocity corresponding to Ci, the numberof counts in the ith channel, N is the number of sweepsmade by the drive during a calibration run, and T is thedwell time per channel.

After converting the data from counts to drive velocity,the data was re-plotted fit to two straight lines, as shownin Figure ??. It is evident from the graph that the drivevelocity does not precisely move through the zero velocitypoint–it has a “blip” that complicates the calibration andthe ensuing data analysis, as will be seen shortly.

For our final fit lines, we made the simplifying assump-tion that the slopes of the fit-lines on either side of thezero velocity point were equal. We averaged the slopesand the x-intercepts of the two lines to obtain the valuesfor the final fit equation. The error in the x-interceptwas estimated as half the difference between the two x-intercepts that were averaged (40 channels), added inquadrature to the error due to the blip in the drive ve-locity (30 channels) for a total error in the x-intercept of50 channels. The error in the slope was obtained from theoriginal linear fits. The final equation obtained was thefollowing: Vi = (0.0011 ± 0.0001)(Ci − 1030 ± 50). This

Fe 57 57Co

HV OutSpectech UCS-20

Multi-Channel Scaler

Multi-Channel Analyzer

Drive MotorDetector

OscilloscopeDrive Circuit

FIG. 4: Schematic of the Mossbauer setup.

equation allowed us to obtain the velocity drive and theenergy corresponding the velocity, and therefore a par-ticular channel, could be calculated from the first orderDoppler shift equation: ∆E = (v/c)E.

3.2. Zeeman Effect

The setup for this portion of this experiment is shownin Figure ??. In this portion of the experiment, we ob-served the effect of Zeeman splitting by observing thedifference between the new transition states and the un-split, single transition state of 14400eV. From this, wecan deduce g0 and g1, the energy spacings of the splitground state and the split excited state, respectively, andthe ratio between µ0 and µ1, the magnetic moments ofthe ground state and the first excited state. Data wasobtained by allowing the gamma ray source to impingeupon the Fe57 absorber as the drive sweeps through ve-locities. The raw data was then fit to a six Lorentzianfunction, and the location of each peak in channel num-ber was noted. The Michelson calibration equation wasthen used to obtain the drive velocity corresponding tothat channel number. The Doppler equation was thenused to calculate the ∆E of the transition. The spac-ings between the split levels, g0 and g1, were calculatedby taking the differences between the ∆E. The ratio be-tween the magnetic moments of the nucleus was then cal-culated with the aid of the equation for g0 and g1 givenin the Theory section above: µ1

µ0= g1I1

g0I0The Zeeman

data was also used as a secondary calibration. Becausethe Michelson calibration was difficult to carry out, weonly performed the absolute calibration once. To scalethe velocity fit to data that was taken on the other driveand at other velocities, we compared the Zeeman datataken under different conditions by measuring the differ-ences between the peaks of the six Lorentzians. We thentook the ratio of each day’s Zeeman data with the orig-inal Zeeman data and obtained a different scaling factorfor each set of drive conditions. The Zeeman data thatwas used for obtaining the reference factor is shown inFigure 5.

4

600 800 1000 1200 1400 1600

60

80

100

120

140

160

180

200

220

240

Zeeman Interaction in Fe57

Channel Number

Cou

nts

χ2ν−1

= 1.1

FIG. 5: Zeeman splitting in Fe57

3.3. Resonant Line Width, Quadrupole Splittingand Isomer Shift

To measure the resonant line width of Fe57, we usedthe absorber Na3FeCn6, which had the advantage thatit has no magnetic or electric field gradient at the crystalsites of the iron nuclei. We measured the line width byfiring the gamma ray source at the absorber and notingthe width of the absorption line that was observed in themultichannel analyzer, using samples of three differentthicknesses: 0.1, 0.25, and 0.5g/cm2. We fit the threeline widths to a line and extrapolated to obtain a valueof the zero absorber thickness line width of Fe57. Fromthe line width obtained, we calculated the lifetime of thefirst excited state of Fe57.

To measure both the quadrupole splitting and isomershift of the 3d6 state of Fe++, Fe(SO4) 7H2O was used asan absorber. The absorber Fe2(SO4)3 was used to mea-sure the isomer shift in Fe+++. No quadrupole splittingwas observed in Fe+++, as there is no field gradient inthe nucleus. The setup was the same as that in the Zee-man splitting portion of this experiment (Figure 4).

4. RESULTS AND ERROR ANALYSIS

4.1. Zeeman and Resonant Line Width

From least to highest energy, the deviations of thesix magnetic sub-levels from the unsplit energy weremeasured to be: (−1.8 ± 0.3) × 10−7eV, (−1.0 ± 0.3) ×10−7, (−2.0±2.6)×10−8, (1.0±0.3)×10−7, (1.7±0.3)×10−7, (2.7 ± 0.3) × 10−7. The separations in the splitlevels were calculated to be (0.8 ± 0.2) × 10−7eV and(2.0± 0.3)× 10−7eV. The theoretical values for these are1.1×10−7 and 1.9×10−7, as obtained from ??–the exper-imental values are both within 2σ of the accepted values,and the second value is within 1σ. The ratio of the mag-netic moments of the first excited state to the groundstate was calculated to be −1.2 ± 0.3, while the theo-retical value is −1.715± 0.004; we have good agreementbetween experimental and accepted values.

We measured the line width at zero absorber thicknessto be 6.8 ± 17.1 × 10−8eV , while the accepted value is4.7 × 10−9eV . The lifetime of the first excited state of

0 0.1 0.2 0.3 0.4 0.5 0.6

−2

−1

0

1

2

3

4x 10

−7Line Width of Fe57 using absorber Na

3FeCn

6

Absorber Thickness (g/cm2)

Line

Wid

th (

eV)

χ2ν−1

= 0.0087

FIG. 6: Resonant line data for absorber at three differentthicknesses, plotted with the fitted line.

iron was calculated to be (6.7±2.6)×10−9seconds, whilethe accepted value is 9.8× 10−8 seconds. The fitted datais shown in Figure 6. The poor reduced chi-squared ofthe fit is testament, again, to the poor calibration andinaccuracies in the introduced scaling factor.

The uncertainty in the final energy width was calcu-lated by first obtaining the error in the calculated ve-locity of the drive, since the energy is proportional tothe drive velocity by the first order Doppler shift equa-tion. The error in the drive velocity is obtained by thefollowing equation, where s is the scaling factor intro-duced by comparing Zeeman data taken under differentdrive conditions, and b is the slope of the final Michelsoncalibration fit-line:

σ2Vi

= V 2i (σ2

b

b2+σ2

s

s2+σ2

Ci+ σ2

a

(Ci − a)2) (1)

We found that most of the error was obtained from theerror in the x-intercept, σa, which is hardly surprising,as there was significant uncertainty in the absolute cali-bration around the zero velocity channel.

4.2. Quadrupole Splitting and Isomer Shift

The quadrupole splitting of the 3d6 state of Fe++ wasmeasured to be (2.1± 0.5)× 10−7eV, while the acceptedvalue is 1.5 × 10−7eV. The isomer shift in Fe++ wasmeasured to be (1.7± 0.5)× 10−7eV, while the acceptedvalue is 6.7×10−8eV, and the isomer shift in Fe+++ wasmeasured to be(8±6)×10−8eV, while the accepted valueis 2.4× 10−8. In doing the data analysis for this portionof the experiment, it was necessary to eliminate part ofthe raw data, as there was unwanted noise generated ythe drive velocity error, as shown in Figure 7, where thecharacteristic of the raw data deviates from a Lorentzian.The error was determined to span an area of (100 ± 30)channels and was cut out of the data. Upon reshiftingthe data and refitting, it was found that the Lorentzianfits now achieved favorable reduced chi-squared valuesthat were close to unity. The isomer shift data sets wereused to estimate the area spanned by the area, as thenoise was easiest to make out in those data sets. The

5

FIG. 7: Raw data for isomer shift in Fe+++ before correction.

FIG. 8: Data for isomer shift after correction.

same area of channels were cut out of both isomer andquadrupole splitting data (Figures 8 and 9.

5. CONCLUSIONS

Using the recoilless method of Mossbauer spectroscopy,we were able to observe the Zeeman splittings in Fe57

and calculate the separations in the energy splittings ofthe ground and first excited state, (0.8 ± 0.2) × 10−7eVand (2.0±0.3)×10−7eV respectively, which compare wellwith the theoretical values, 1.1× 10−7 and 1.9× 10−7 re-spectively. The ratio of the nuclear magnetic momentsof the first excited state and the ground state, µ1

µ0, was

calculated to be −1.2 ± 0.3, while the value from liter-ature [1] is −1.715 ± 0.004–again, there is good agree-ment between experiment and theory. The quadrupole

splitting of the 3d6 state of Fe++ was measured to be

FIG. 9: Data for combined quadrupole effects in Fe++ aftercorrection.

(2.1 ± 0.5)× 10−7eV which is within 2σ of the acceptedvalue, 1.5× 10−7eV. The isomer shift in Fe++ was mea-sured to be (1.7±0.5)×10−7eV, while the accepted valueis 6.7 × 10−8eV. The isomer shift in Fe+++ was mea-sured to be(8± 6)× 10−8eV, while the accepted value is2.4 × 10−8. The resonant line width data showed pooragreement between experiment and theory. The reso-nant line width was measured to be 6.8± 17.1× 10−8eV ,while the accepted value is 4.7 × 10−9eV . The lifetimeof the first excited state of iron-57 was calculated to be(6.7 ± 2.6) × 10−9seconds, while the accepted value is9.8× 10−8 seconds. The discrepancy is large–over an or-der of magnitude–and is due to the inaccurate Michelsoncalibration and, to a lesser degree, inaccuracies in the sec-ondary Zeeman calibration. One unaccounted error wasthe error in the scaling factor c, which varied by 10%between days. There is also error involved in the convo-lution between the source and absorber–this is due to thefact that the source and absorber are not point objects,but have a finite area. This would lead to line broadeningin the data collected. In order to improve accuracy in thisexperiment, it is imperative that the absolute calibrationbe improved. This could be done by taking the Michelsoncalibration under every different drive condition used tocollect data. Also, another possibility is making improve-ments to the drive velocity characteristic and eliminatingthe “blip”.

[1] H. et. al, Mossbauer effect in metallic iron (1962).[2] R. L. Moessbauer, Nobel lecture: Recoilless nuclear reso-

nance absorption of gamma radiation (1961).[3] J. King, Mossbauer effect: Selected reprints (1963).[4] D. J. Bland, Electric quadrupole splitting, URL

http://www.cmp.liv.ac.uk/frink/thesis/thesis/

node18.html.[5] Boyle and Hall, The mossbauer effect (reports on the

progress of physics).

Acknowledgments

The author would like to acknowledge Kelley Rivoire,her fellow investigator, and the junior lab staff for theirassistance in lab.